Blood Pressure and Arterial Compliance Estimation from Arterial Segments

ABSTRACT

A noninvasive method for monitoring the blood pressure and arterial compliance of a patient based on measurements of a flow velocity and a pulse wave velocity is described. An embodiment uses a photoplethysmograph and includes a method to monitor the dynamic behavior of the arterial blood flow, coupled with a hemodynamic mathematical model of the arterial blood flow motion in a fully nonlinear vessel. A derived mathematical model creates the patient specific dependence of a blood pressure versus PWV and blood velocity, which allows continuous monitoring of arterial blood pressure.

CROSS REFERENCE

This application claims the benefit of the filing date of U.S.Provisional Patent Application Ser. No. 62/080,740, filed Nov. 17, 2014,and U.S. Provisional Patent Application Ser. No. 62/080,738, filed Nov.17, 2014, each of which are hereby incorporated by reference in theirentirety.

FIELD

The disclosure relates to methods for at least one of blood pressure andarterial compliance estimation from arterial segments.

BACKGROUND

The pulse wave, generated by left ventricular ejection, propagates at avelocity that has been identified as an important marker ofatherosclerosis and cardiovascular risk. Increased pulse wave velocity(PWV) indicates an increased risk of stroke and coronary heart disease.This velocity is considered a surrogate marker for arterial compliance,is highly reproducible, and is widely used to assess the elasticproperties of the arterial tree. Research shows that measurement ofpulse wave velocity as an indirect estimate of aortic compliance couldallow for early identification of patients at risk for cardiovasculardisease. The ability to identify these patients would lead to betterrisk stratification and earlier, more cost-effective preventativetherapy. Several studies have shown the influence of blood pressure andleft ventricular ejection time (LVET) on pulse wave velocity.

Over the past decades, there has been ongoing research for bettertheoretical prediction of PWV. The clinical relationship between PWV andarterial stiffness is often based on classic linear models or thecombination of the linear models, and measured results with anincorporated correction factor. Whereas linear models predict PWV as afunction of only geometric and physical properties of the fluid and thewall, there is strong empirical evidence that PWV is also correlated topressure and ejection time.

There exist no models that accurately describe PWV and flow accountingfor the nonlinearities in an arterial segment. A model that would enablesolution of the inverse problem of determination of blood pressure andaortic compliance for a PWV measure has yet to be developed.

SUMMARY

In accordance with one aspect of the present invention, there isprovided a method for determining material characteristics for anartery, the method including providing at least three values for each ofblood pressure, internal radius, and external radius of an arterialsegment or segments of a subject; applying a base model offluid-structure interaction incorporating conservation of mass andmomentum for the fluid, and non-linear elasticity of the structure; andrunning a mathematical optimization on the base model to provide acalibrated model to determine material characteristics of the arterialsegment or segments.

In accordance with another aspect of the present invention, there isprovided a method for determining at least one of a blood pressure andan arterial compliance parameter of a subject, the method includingproviding a value for pulse wave velocity within an arterial segment orsegments of a subject; providing a value for flow velocity within thearterial segment or segments of the subject; providing a value for blooddensity of the subject; providing the material characteristics of anartery; and applying a calibrated model of fluid-structure interactionincorporating conservation of mass and momentum for the fluid, andnon-linear elasticity of the structure, to calculate at least one ofblood pressure and an arterial compliance parameter of the subject usingthe provided values.

These and other aspects of the present disclosure will become apparentupon a review of the following detailed description and the claimsappended thereto.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram on the longitudinal cross section of the arterialwall at diastolic (10) and systolic (11) pressure;

FIG. 2 is a graph of the dependence of distensibilty on transmuralpressure and a flow velocity;

FIG. 3 is a flowchart depicting creation of a blood pressure anddistensibility lookup table;

FIG. 4 is a format of a lookup table for blood pressure anddistensibility;

FIG. 5 is a flowchart depicting a method for determining blood pressureof a subject based on a PWV measure using a lookup table;

FIG. 6 is a flowchart depicting a method for determining arterialcompliance of a subject based on a PWV measure using a lookup table;

FIG. 7 is an illustration of pressure pulse measurement on two locationsin the same artery for determination of a systolic and diastolic PWV;and

FIG. 8 is a flowchart depicting a method for determining the materialcharacteristics of an arterial segment.

DETAILED DESCRIPTION

A fully nonlinear model of pressure and flow propagation in arterialsegments is disclosed that enables determination of blood pressure andarterial compliance based on measures of pulse wave velocity and flowvelocity following a calibration. The approach allows for determinationof systolic and diastolic blood pressure. The approach allows fordetermination of a systolic and diastolic aortic compliance.

A noninvasive method for monitoring the blood pressure and arterialcompliance of a patient based on measurements of a flow velocity and apulse wave velocity is described. An embodiment uses aphotoplethysmograph and includes a method to monitor the dynamicbehavior of the arterial blood flow, coupled with a hemodynamicmathematical model of the arterial blood flow motion in a fullynonlinear vessel. A derived mathematical model creates the patientspecific dependence of a blood pressure versus PWV and blood flowvelocity, which allows continuous monitoring of arterial blood pressure.The calibrated mathematical model presents an arterial compliance and adistensibility as a clinical marker of arterial stiffness. Thedisclosure is applicable for fully nonlinear elastic vessels that arecommonly found in the major arteries, as well as smaller vessels thatoperate closer to the linear elastic regime.

The disclosure includes a fully nonlinear basic model for blood pressurewave propagation in compliant arteries. A nonlinear traveling wave modelwas used to investigate mechanisms underlying the effects of pressure,ejection time, ejection volume, geometric, and physical properties onPWV. A patient calibration procedure was developed that involvesmeasurement of blood pressure and arterial dimensions (internal andexternal radii). An embodiment includes blood pressure prediction usingthe model, per patient calibration, and the measurement of flow velocityand pressure wave velocity. An embodiment includes arterial compliancedetermination using the model, per patient calibration, and themeasurement of flow velocity and pressure wave velocity.

A basic mathematical fluid-structure interaction model for pulse wavevelocity (PWV) propagation incorporates the dynamics of incompressibleflow in a compliant vessel. This one dimensional model simulating bloodflow in arteries effectively describes pulsatile flow in terms ofaverages across the section flow parameters. Although it is not able toprovide the details of flow separation, recirculation, or shear stressanalysis, it accurately represents the overall and average pulsatileflow characteristics, particularly PWV.

Conservation of mass and momentum results in the following system of onedimensional equations

$\begin{matrix}{{\frac{\partial A}{\partial t} + {\frac{\partial}{\partial z}({uA})}} = 0} & (1) \\{{\frac{\partial u}{\partial t} + {\frac{\partial}{\partial z}\left( {\frac{u^{2}}{2} + \frac{p}{\rho}} \right)}} = 0} & (2)\end{matrix}$

where t is time, z is the axial coordinate shown in FIG. 1, A=A (z,t) isthe arterial cross-sectional area, u=u(z,t) is the blood flow velocity,ρ is blood density, p=p(z,t) is blood pressure.

For an impermeable thin walled membrane, neglecting inertia forces, thevessel pressure-strain relationship is maintained by equilibriumcondition as a function p=p(η), based on relevant constitutive relationswhere 11 is the circumferential strain (ratio of wall deflection to zerostress arterial radius (R)). Noting that A=πR²(1+η)², and assuming thattransmural pressure is a smooth function of a wall normal deflection(derivative p_(η)=∂p/∂η exists at any point), the total system ofequations can be presented in the following non-conservative form

$\begin{matrix}{{\frac{\partial U}{\partial t} + {{H(U)}\frac{\partial U}{\partial z}}} = 0} & (3) \\{where} & \; \\{{U = \begin{bmatrix}\eta \\u\end{bmatrix}};{H = \begin{bmatrix}u & \frac{1 + \eta}{2} \\\frac{p_{\eta}}{\rho} & u\end{bmatrix}}} & (4)\end{matrix}$

We find the eigenvalues of H(U) to be real and distinct. PWV isassociated with the forward running wave velocity, i.e., the largesteigenvalue, hence it is identified as

$\begin{matrix}{{PWV} = {u + \sqrt{\frac{1 + \eta}{2\rho}p_{\eta}}}} & (5)\end{matrix}$

The partial derivative p_(ii) indicates sensitivity of pressure withrespect to the wall normal deflection, and has a clear interpretation astangent (incremental) moduli in finite strain deformation. In thegeneral case, equation (5) is supplemented by appropriate constituentequations for a hyperelastic anisotropic arterial wall, accounting forfinite deformation.

It is assumed that arterial wall is hyperelastic, incompressible,anisotropic, and undergoing finite deformation. After a few originalloading cycles (preconditioning) the arterial behavior follows somerepeatable, hysteresis free pattern with a typical exponentialstiffening effect regarded as pseudo elastic. The strain energy densityfunction W for the pseudo elastic constitutive relation may be presentedin the form

W=½c(e ^(Q)−1)  (6)

where c is a material coefficient, and Q is the quadratic function ofthe Green-Lagrange strain components. For the finite inflation andextension of a thin walled cylindrical artery the following strainenergy function is used

Q=a ₁₁ E _(θ) ²+2a ₁₂ E _(θ) E _(z) +a ₂₂ E _(z) ²  (7)

where c, a₁₁, a₁₂, a₂₂ are material constants. The Cauchy stresscomponents in circumferential and axial directions are:

$\begin{matrix}\begin{matrix}{{\sigma_{\theta} = {{\lambda_{\theta}^{2}\frac{\partial W}{\partial E_{\theta}}} = {c\; \lambda_{\theta}^{2}e^{Q}s_{\theta}}}},} & {s_{\theta} = {{a_{11}E_{\theta}} + {a_{12}E_{z}}}} \\{{\sigma_{z} = {{\lambda_{z}^{2}\frac{\partial W}{\partial E_{z}}} = {c\; \lambda_{z}^{2}e^{Q}s_{z}}}},} & {s_{z} = {{a_{12}E_{\theta}} + {a_{22}E_{z}}}}\end{matrix} & (8)\end{matrix}$

With the geometry of the reference state determined, we define R, Z,H—as an internal radius, axial coordinate and a wall thickness in astress free configuration, r,z,h—internal radius, axial coordinate and awall thickness in a physiologically loaded configuration. Thecorresponding principal stretch ratios are

λ_(θ) =r/R,λ _(z) =dz/dZ,λ _(r) =h/H  (9)

Assuming isochoric deformation incorporate the incompressibilitycondition as

λ_(z)λ_(θ)λ_(r)=1  (10)

The Green-Lagrangian strain components relate to the principal stretchratios of Eq. 12 by

E _(i)=½(λ_(i) ²−1), (i=θ, z, r)  (11)

For the membrane thin walled cylindrical artery undergoing finiteinflation and axial deformation, the load-stress relations follow fromthe static conditions

$\begin{matrix}{{\sigma_{\theta} = {\frac{pr}{h} = {\frac{{pR}\; \lambda_{\theta}}{H\; \lambda_{r}} = {\frac{pR}{H}\lambda_{\theta}^{2}\lambda_{z}}}}}{\sigma_{z} = {{\frac{F}{2\; \pi \; {RH}}\lambda_{z}} = {f\; \lambda_{z}}}}} & (12)\end{matrix}$

where F is the axial pretension force and f is the axial pre-stress perunit of cross section area of a load free vessel. A substitution backinto equation (8) yields the desired relations:

$\begin{matrix}{{{\lambda_{z}^{- 1}{ce}^{Q}s_{\theta}} = \frac{pR}{H}}{{c\; \lambda_{z}e^{Q}s_{z}} = f}} & (13)\end{matrix}$

The solution of equations (13), (11), (7) results in a load-strainrelations, which with account of the identity

$\lambda_{\theta} = {\frac{r}{R} = {{\frac{r - R}{R} + 1} = {\eta + 1}}}$

converts into the p=p(η) function, required by (5) to predict a wavefront speed of propagation, i.e., PWV.

Arterial stiffness, or its reciprocals, arterial compliance anddistensibility, may provide indication of vascular changes thatpredispose to the development of major vascular disease. In an isolatedarterial segment filled with a moving fluid, compliance is defined as achange of a volume V for a given change of a pressure, and distensibiltyas a compliance divided by initial volume. As functions of pressure thelocal (tangent) compliance C and distensibility D are defined as

$\begin{matrix}{{C = \frac{dV}{dp}},{D = {\frac{C}{V} = \frac{dV}{Vdp}}}} & (14)\end{matrix}$

Equations (14) determine arterial wall properties as local functions oftransmural pressure.

We present equations (14) in the following equivalent form

$\begin{matrix}{{C = {\frac{{{dV}/d}\; \eta}{{{dp}/d}\; \eta} = \frac{2\; V}{p_{\eta}}}},{D = {\frac{C}{V} = \frac{2}{p_{\eta}}}}} & (15)\end{matrix}$

The classical results are generalized for the case of a hyperelasticarterial wall with account of finite deformation and flow velocity. Toproceed, determine p_(η) from Equation (5) and substitute in Equation(15), arriving at the following relations

$\begin{matrix}{{D = \frac{1 + \eta}{{\rho \left( {{PWV} - u} \right)}^{2}}},\mspace{14mu} {C = {VD}}} & (16)\end{matrix}$

FIG. 2 illustrates the dependency of distensibility on pressure and flowvelocity. Since PWV is monotonically increasing with pressure,distensibility is a decreasing function. Unlike the classicalBramwell-Hill model, which being linked to the Moens-Korteweg wave speedpredicts arterial distensibility as a constant irrespective to thepressure level, the present model predicts distensibility as a functionof PWV, pressure and a blood flow.

Arterial constants can be defined based on the developed mathematicalmodel. The Cauchy stress components based on Fung's energy are presentedin equations (6), (7), and (8). Neglecting longitudinal stress (σ=0) inequation (8), we obtain

$\begin{matrix}{E_{z} = {{- \frac{a_{12}}{a_{22}}}E_{\theta}}} & (17)\end{matrix}$

where the ratio

$\frac{a_{12}}{a_{22}}$

is a counterpart of a Poisson coefficient in a linear isotropicelasticity.

It follows from Equations (7), (17) that a circumferential stress is afunction of a circumferential strain and two material constants, a and c

$\begin{matrix}{\sigma_{\theta} = {{ac}\; \lambda_{\theta}^{2}e^{{aE}_{\theta}^{2}}E_{\theta}}} & (18) \\{Q = {aE}_{\theta}^{2}} & (19) \\{a = {a_{11} - \frac{a_{12}^{2}}{a_{22}}}} & (20)\end{matrix}$

The governing equation specifies an equilibrium condition

$\begin{matrix}{{\sigma_{\theta} = {{{ac}\; \lambda_{\theta}^{2}e^{{aE}_{\theta}^{2}}E_{\theta}} = \frac{{pr}_{i}}{h}}}{{h = {r_{o} - r_{i}}};\mspace{14mu} {r_{m} = \frac{r_{i} + r_{o}}{2}}}{{\lambda_{\theta} = \frac{r_{m}}{R_{m}}};\mspace{14mu} {E_{\theta} = \frac{\lambda_{\theta}^{2} - 1}{2}}}} & (21)\end{matrix}$

where p is the transmural pressure, r_(i) is the internal radius, r_(o)is the outer radius, and h is the wall thickness. Let us define r_(m) asthe mid radius of a loaded vessel, and R_(m) as the mid radius for astress free vessel. Measuring r_(i) and r_(o) corresponding to thepressure p leaves us with three unknowns (a, c, R_(m)) which need to bedetermined as a part of a calibration procedure. The calibrationprovides a calibrated model which can be individualized for eachsubject.

An embodiment of the calibration includes the use of published valuesfor a population or segment of a population. Referenced values formaterial constants c, a₁₁, a₁₂, a₂₂ can be used to calculate thematerial constant a based on equation (20). The material constant c isused directly. A reference value for the mid radius in the stress freestate (R_(m)) is used along with a reference value for the arterial wallthickness (h) and associated mid radius (r_(m)) for the loaded wall. Thematerial parameters (a, c, R_(m)) along with the product of wallthickness and mid radius for the loaded wall (hr_(m)) can then be usedto determine at least one of blood pressure and arterial compliance,e.g., distensibility, of a subject by measuring PWV and flow velocity.

Another embodiment of the calibration is disclosed as follows anddescribed in FIG. 8. Assuming we have k measurements of the radius andpressure during a cardiac cycle or cycles, the following four variablesare defined (r_(i) _(k) , r_(o) _(k) , r_(m) _(k) , p_(k)). By usingthese sampled variables circumferential stress can be presented as afunction of three unknowns

σ_(θ)=σ_(θ)(a, c, R _(m))  (22)

Now using a mathematical optimization, e.g., a least square (LS)minimization technique identifies (a, c,R_(m))

$\begin{matrix}{{LS} = {{\sum_{k}\left\lbrack {{\sigma_{\theta}\left( {a,c,R_{m}} \right)} - \frac{p_{k}{ri}_{k}}{h_{k}}} \right\rbrack^{2}}\underset{\min}{\rightarrow}\left( {a,c,R_{m}} \right)}} & (23)\end{matrix}$

The following nonlinear calibration method describes one approach thatmay be completed to determine arterial constants.

Step 1: Obtain k measurements, k≧3, for blood pressure-p_(k), internalradius-r_(ik); outer radius-r_(ok), calculate mean radiir_(mk)=0.5(r_(ik)+r_(ok)) and wall thicknesses h_(k)=r_(ok−)r_(ik).example, tonometry could be used to measure a continuous blood pressureto create the array of blood pressures, and Doppler speckle ultrasoundcould be used to measure artery radii to create the corresponding arrayof r_(ik), r_(ok).

Step 2: Run a least square minimization as in equation (23) to identifythe three constants (two material constants a, c and the mean radiusR_(m), in a load free condition). Substituting σ_(θ) in equation (23)with equation (18) results in

$\begin{matrix}{{LS} = {{\sum_{k}\left\lbrack {{{ac}\; \lambda_{\theta \; k}^{2}e^{{aE}_{\theta \; k}^{2}}E_{\theta \; k}} - \frac{p_{k}{ri}_{k}}{h_{k}}} \right\rbrack^{2}}\underset{\min}{\rightarrow}\left( {a,c,R_{m}} \right)}} & (24)\end{matrix}$

where

${\lambda_{\theta \; k} = \frac{r_{mk}}{R_{m}}};{E_{\theta \; k} = {\frac{\lambda_{\theta \; k}^{2} - 1}{2}.}}$

LS is a function of measured parameters p_(k,) r_(ik,) r_(mk,) h_(k) andunknown properties (a,c,R_(m)), determined from the minimizationprocedure. Since we have 3 unknowns, at least 3 sets of pressure andassociated outer and inner radii are required.

In an embodiment, a calibrated model can be used to determine at leastone of blood pressure and arterial compliance, e.g., distensibility, ofa subject by measuring PWV and flow velocity.

With the three material properties (a,c,R_(m)) and the constant product(hr_(m)) a blood pressure may be estimated based on the equilibriumequation (21) rearranged to

$\begin{matrix}{p = \frac{{ac}\; \lambda_{\theta}^{2}e^{{aE}_{\theta}^{2}}E_{\theta}h}{r_{i}}} & (25)\end{matrix}$

where the stretch ratio (λ_(θ)) can be defined in terms of η

$\begin{matrix}{\lambda_{\theta} = {\frac{r_{m}}{R_{m}} = {\left( {\frac{r_{m} - R_{m}}{R_{m}} + 1} \right) = {\eta + 1}}}} & (26)\end{matrix}$

The circumferential strain (E_(θ)) can be defined in terms of η

E _(θ)=(λ_(θ) ²−1)/2=η(η+2)/2  (27)

Wall thickness follows from incompressibility conditions

hr_(m)=h_(k)r_(mk)  (28)

where from

$\begin{matrix}{h = {\frac{h_{k}r_{mk}}{r_{m}} = {\frac{h_{k}r_{mk}}{R_{m}\lambda_{\theta}} = \frac{h_{k}r_{mk}}{R_{m}\left( {\eta + 1} \right)}}}} & (29)\end{matrix}$

Internal Radius

r _(i) =r _(m)−0.5h=λ _(θ) R _(m)−0.5h=(η+1)R _(m)−0.5h  (30)

These formulations provide a relationship between pressure (p),circumferential strain (η), two arterial material parameters (a and c)and two arterial geometric parameters (R_(m) and the constant producth_(k)r_(mk)).

Equation (5) for PWV in arterial tissues can be re-arranged

p _(η)(1+η)=2ρ*PWV_(f) ²  (31)

where a flow corrected PWV (PWV_(f)) has been introduced(PWV_(f)=PWV-u).

This relation can be used in combination with equation (16) to definedistensibility D in terms of the p_(η)

D= ²/_(p) _(η)   (32)

In an embodiment, a lookup table can be created for convenience toenable determination of a blood pressure and distensibility based on the4 arterial parameters (can be subject specific) and measurement of PWVand flow velocity. A blood density ρ is either measured or a value isassumed based on age and gender. As illustrated in an embodiment shownin FIG. 3, the steps to create the blood pressure and distensibilitylookup table are as follows:

Set η=0.

Using equation (26) calculate circumferential stretch ratio λ_(θ).

Using equation (27) calculate circumferential strain E_(θ).

Using equation (29) and any h_(k)r_(mk) product from the calibration,calculate wall thickness h.

Using equation (30) calculate the internal radius r_(i).

Using equation (25) calculate pressure p.

While η<0.5, η=η+0.005, go back to calculation of the circumferentialstretch ratio or else continue.

Calculate the array for p_(n) using the slope of the (p,η) curve.

Using equation (31) and the array of values for η and p_(n) calculate a1D array for p_(η)(1+η).

Using equation (31) and the array of p_(η), calculate D for each valueof η.

An example resultant array is shown in FIG. 4. This lookup table enablesdetermination of a subject blood pressure and distensibility bymeasuring PWV_(f) and identifying the row (either directly or byinterpolation) where p_(η)(1+η)=2ρ*PWV_(f) ². This row also containsother relevant arterial parameters such as E_(θ), λ_(θ), h, and r, whichcan be extracted for that individual based on the measured PWV_(f). Notethat for determination of a systolic blood pressure and distensibility,a systolic PWV and a maximum flow velocity is used to calculate PWV_(f).For determination of a diastolic blood pressure and distensibility, adiastolic PWV and a minimum (or zero) flow velocity is used to calculatePWV_(f). Intermediate values of blood pressure and distensibility mayalso be determined based on the associated intermediate values ofPWV_(f) and flow velocity.

A method for monitoring a blood pressure of a subject is disclosed. Themodel is calibrated for the subject (or population) by a non-linearcalibration. For example, a non-linear calibration as illustrated inFIG. 3 can be performed. Once the calibration is complete, a patientspecific lookup table can be formed for convenience as described in FIG.3 and as shown in FIG. 4. The method for monitoring a subject's realtime blood pressure is shown in FIG. 5. An ECG measurement can be madefor use as a timing reference for start of the pulse wave transit time,or to be used as a reference for determining acceptance time windows forother waveform features, or for averaging waveforms. An arterial flowvelocity is measured providing minimum and maximum flow velocities,although a minimum could be assumed to be zero. An average flow velocitycan also be measured. The flow velocity can also be estimated based onother measures or as a percentage of PWV. The pulse wave velocity ismeasured, ideally providing both a systolic and diastolic PWV. Thesubject (or population based) blood density ρ are then used with thelookup table of FIG. 4 to estimate blood pressure. The arrays ofassociated blood pressure (p), PWV_(f) and distensibility (D) can alsobe used directly or with interpolation to estimate p and D based on thePWV_(f) measure. The systolic PWV and the peak flow velocity are used toestimate a systolic blood pressure, while a diastolic PWV and theminimum flow velocity are used to estimate a diastolic blood pressure.Although other estimates and combinations may be used to estimate anaverage or systolic or diastolic blood pressure. PWV can be measured atthe foot and the systolic flow velocity (u) can be added to it as anestimate of systolic PWV_(f). Diastolic pressure could use the raw PWVmeasured at the foot and use either the minimum flow velocity or assumeu=0. Other empirical or relational estimates for the systolic anddiastolic PWV_(f) can also be used.

Determination of PWV includes measurement of the transit time of thepulse wave between two points, and a measure or estimate of the distancetraveled. The PWV is the distance travelled divided by the timedifference. This can be done by extracting the foot (FIG. 1, item 12) orpeak (FIG. 1, item 13) of the pressure wave in two locations (proximal,distal) of the same artery, calculating the time difference betweenthese two extracted features, and measuring the distance between thesetwo measurement points. In one embodiment, this is done in the radialartery as shown in FIG. 7 with measurements items 80 (proximal) and 81(distal). Use of the foot location on the pressure or PPG waveform(items 801 and 810) will correlate to a diastolic PWV, while use of thepeak location (items 802 and 811) will correlate to a systolic PWV. Twodifferent arteries can also be used with a measurement or estimate ofthe arterial path distance between the two measurement points. The PWVcan also be measured using the heart as the proximal measurement pointwith an electrocardiogram feature (e.g., the ECG r-wave) as the firsttime point, or by sensing when the aortic valve opens using a feature onmeasured waveforms or images such as the ballistocardiogram (BCG),ultrasound imaging, Doppler ultrasound, impedance plethysmography (IPG),or photoplethysmography (PPG) on the chest over the aorta. The distalpressure wave is then used as above to extract a second time point. Thearterial distance between the aortic root and the distal measurementpoint is used in the calculation of PWV. This can be measured orestimated, and may be based on subject characteristics such as height,weight age and gender. The distal pressure wave can be measured usingsensors such as tonometry, an arterial cuff, ultrasound, RF basedarterial wall tracking, and PPG.

In some embodiments, the wave will propagate across multiple arterialsegments between the proximal and distal points of pressure measurement.This measurement can be used in at least two ways. In the first form,average properties of the vessel segments, radius, and modulus will beconsidered so that the result corresponds to bulk average of thesegment. In the second, the properties of individual arterial segmentsare determined. First, use the model to determine the relative transittime through each sequential arterial segment based on geometricalproperties of each segment and assuming a similar pressure within allsegments. Then using a solution method, such as minimization of a leastsquares or another method, solve for the PWV within each segment byrecognizing that the total transit time (measured) is the sum of thetransit time through each segment.

Measurement of flow velocity can be done using Doppler ultrasound, aninductive coil, MRI or CT scan with contrast agents. The flow velocitycan be captured as a continuous wave, as a peak value, or a minimumvalue (including u=0). It is also possible to estimate flow velocityusing related measures or with a scale factor. For example PWV can bemeasured using previously described techniques and flow velocity is thenestimated as a percentage of PWV (for example u=0.2 PWV). Aortic flowvelocity can be estimated through left ventricular ejection time (LVET),ejection volume (EV), and aortic cross-sectional area (CA) whereu=EV/(LVET*CA). Left ventricular ejection time (LVET) can be measured orestimated using a number of sensors (e.g. PPG, heart sound). Using PPGfor example, the measure of LVET is the length of time from the foot ofthe PPG wave (FIG. 1 item 12) to the dicrotic notch (item 14). Usingheart sound, LVET is the time between the first and second heart sound.Ejection volume can be based on direct measurement for the subject (e.g.ultrasound, thermal dilution, etc) at rest and at exercise withsubsequent scaling based on heart rate. The cross-sectional area can bedirectly measured by ultrasound imaging, MRI, or CT scans. EV and CA canalso be based on subject specific parameters such as age, gender, heightand weight. EV can also be measured or estimated using features from theBCG such as the amplitude of the j-wave or m-wave. An estimate of flowvelocity in the periphery can be made based on scaling of the bloodvolume in that arterial tree branch, and relative arterial size ascompared to the aorta. Although a direct measurement of flow velocity oran estimate based on PWV is preferred in the periphery.

Pressure can be measured using any approved technique, for example,brachial cuff, tonometry, or intra-arterial catheter. Ideally acontinuous method (e.g., tonometry, intra-arterial) is used with amethod of time synchronization to the flow and PWV measures (e.g. viaECG). However serial measures can also be used. Here the pressure p canbe systolic, diastolic, or any intermediate pressure (e.g., meanpressure) when coupled with the appropriate flow velocity (u). Forexample, systolic pressure could be associated with the peak flowvelocity, and diastolic pressure could be associated with the lowest (orzero) flow velocity, or an average pressure could be associated with anaverage flow velocity.

An optional ECG can be measured across the chest or wrists. Otherlocations are also possible such as ear lobes, behind the ears,buttocks, thighs, fingers, feet or toes. PPG can be measured at thechest or wrist. Other locations such as the ear lobes, fingers,forehead, buttocks, thighs, and toes also work. Video analysis methodsexamining changes in skin color can also be used to obtain a PPGwaveform. Flow velocity can be measured at the chest or wrist. Otherlocations for flow velocity measure are also possible such as the neck,arm and legs.

In one embodiment, the pulse transit time is measured based on aorticvalve opening determined by the J-wave of the BCG waveform, and a PPGfoot measured (e.g., FIG. 1, item 12) from the thigh. The ECG r-wave maybe used as a reference for determining acceptance windows for BCG andPPG feature delineations, or as a starting point for a PWV measure. Theaorta distance is estimated from aortic root along the path of the aortato the femoral artery at the thigh PPG measurement location. The PWV iscalculated by dividing the aorta distance by the measured timedifference (BCG J-wave to PPG foot). The minimum and maximum flowvelocity is measured by ultrasound Doppler at the aortic root. A blooddensity ρ is assumed based on age and gender of the subject. Using themeasured PWV_(f) as shown in FIG. 5 the lookup table of FIG. 4 is usedto determine the associated pressure. To calculate p_(d) the minimummeasured flow velocity can be used in combination with an estimate ofdiastolic PWV, where diastolic PWV is calculated based on the BCG J-waveto PPG foot time difference. To calculate p_(s) the peak flow velocityor a percentage of the peak flow velocity can be used in combinationwith an estimate of systolic PWV, where the systolic PWV is calculatedbased on the peak flow velocity time point to the PPG peak timedifference. Under conditions where flow velocity is not measured,diastolic flow velocity can be assumed to be 0, while systolic flowvelocity can be estimated as a percentage of PWV (e.g ˜20%).

In another embodiment, the pulse transit time is measured from thecarotid artery using tonometry, to the pressure pulse measured at thighwith a thigh cuff. The arterial distance is estimated from aortic rootalong the path of the aorta to the femoral artery at the thigh cuffmeasurement location. The PWV is calculated by dividing the arterialdistance by the measured time difference. The foot to foot timing on themeasured pressure pulses (e.g., FIG. 7 items 801 and 810) is used todetermine a diastolic pulse transit time and to calculate a diastolicPWV. The peak to peak timing (e.g., FIG. 7, items 802 and 811) is usedto determine a systolic pulse transit time and to calculate a systolicPWV. The peak flow velocity is estimated at 20% of the systolic PWVwhile the minimum flow velocity is estimated at zero. A blood density ρis assumed based on age and gender of the subject. Using the measuredPWV_(f), as shown in FIG. 5, the lookup table of FIG. 4 is used todetermine the associated pressure. To calculate p_(d) the minimummeasured flow velocity can be used in combination with the diastolicPWV. To calculate p_(s) the peak flow velocity is used in combinationwith the systolic PWV.

A method for monitoring an arterial compliance of a subject isdisclosed. The model is calibrated for the subject (or population) by anon-linear calibration. For example, a non-linear calibration asillustrated in FIG. 8 can be performed. Once the calibration iscomplete, a patient specific lookup table can be formed for conveniencedescribed in FIG. 3 and as shown in FIG. 4. The method for monitoring asubject's arterial compliance is shown in FIG. 6. An optional ECGmeasurement can be made for use as a timing reference for start of thepulse wave transit time, or to be used as a reference for determiningacceptance time windows for other waveform features, or for averagingwaveforms. An arterial flow velocity is measured providing minimum andmaximum flow velocities, although a minimum could be assumed to be zero.An average flow velocity can also be measured. The flow velocity canalso be estimated based on other measures or as a percentage of PWV. Thepulse wave velocity is measured, ideally providing both a systolic anddiastolic PWV. The subject (or population based) blood density ρ is thenused with the patient specific lookup FIG. 4, to determine compliance asshown in FIG. 6 and equation 16.

The systolic PWV and the peak flow velocity are used to determine asystolic distensibility, while a diastolic PWV and the minimum flowvelocity are used to estimate a diastolic distensibility. Although otherestimates and combinations may be used to determine the subjectdistensibility parameter.

In one embodiment, the pulse transit time is measured based on aorticvalve opening determined by the J-wave of the BCG waveform, and a PPGfoot measured (e.g., FIG. 1, item 12) from the thigh. The aorta distanceis estimated from aortic root to femoral artery at the thigh PPGmeasurement location. The PWV is calculated by dividing the aortadistance by the measured time difference (BCG J-wave to PPG foot). Theminimum flow velocity is assumed to be zero, enabling distensibilitycalculation without a direct flow measurement. The flow corrected PWV iscalculated (in this embodiment PWV_(f)=PWV since u=0). A blood density ρis assumed based on age and gender of the subject with the patientspecific lookup FIG. 4 to determine compliance as shown in FIG. 6 andequation 16. Measures of a systolic PWV, peak flow velocity, may also beused to determine D.

In another embodiment, the pulse transit time is measured from thecarotid artery using tonometry, to the pressure pulse measured at thighwith a thigh cuff. The arterial distance is estimated from aortic rootalong the path of the aorta to the femoral artery at the thigh cuffmeasurement location. The PWV is calculated by dividing the arterialdistance by the measured time difference. The foot to foot timing on themeasured pressure pulses (e.g., FIG. 8 items 801 and 810) are used todetermine a diastolic pulse transit time and to calculate a diastolicPWV. The peak to peak timing (e.g., FIG. 7, items 802 and 811) is usedto determine a systolic pulse transit time and to calculate a systolicPWV. The peak flow velocity is estimated at 20% of the systolic PWVwhile the minimum flow velocity is estimated at zero. A blood density pis assumed based on age and gender of the subject with the patientspecific lookup FIG. 4 to determine compliance as shown in FIG. 6 andequation 16.

The disclosure will be further illustrated with reference to thefollowing specific examples. It is understood that these examples aregiven by way of illustration and are not meant to limit the disclosureor the claims to follow.

EXAMPLE

Paper Example, Blood Pressure and Distensibility

This paper example uses referenced values for an aorta c=120123 Pa,a₁₁=0.320, a₁₂=0.068, a₂₂=0.451. The reduced constant ‘a’ is calculatedbased on equation 20 (a=0.31). In addition reference values were usedfor the mid radius for the stress free vessel R_(m)=0.009 m, the aorticwall thickness h=0.00211 m, and the mid radius of the loaded wallr_(m)=0.011. A computer routine executed the functions outlined in FIG.3 to calculate an array of p_(η)(1+η), D, and the associated pressures(p) for each value of η. Equation (31) was then used along with areference value of blood density ρ=060 kg/m³ to create an array ofassociated PWV_(f) for each value of η. The pulse transit time ismeasured based on time synchronized tonometry at the carotid artery andat the femoral artery, each providing a pressure pulse waveform. Theequivalent aortic distance between the two measurement points ismeasured at for example 1.0 m. The foot to foot time difference betweenthe two tonometry waveforms is measured at for example 200 nis. Thediastolic PWV is then calculated as 1.0 m/200 ms=0.5 m/s. From the sametonometry waveforms the peak to peak time difference is measured at forexample 140 ms. The systolic PWV is then calculated as 1.00 m/140ms=7.14 m/s. Flow velocity is measured using Doppler ultrasound at theaortic root, with the minimum taken as the diastolic flow velocityu_(d)=0.00 ^(m)/_(s), and the maximum taken as the systolic flowvelocity u_(s)=1.10 ^(m)/_(s). A flow corrected PWV is calculate atsystolic blood pressure (PWV_(f)=7.14 m/s−1.10 m/s=6.04 m/s) with theassociated systolic blood pressure determined from the array asp_(s)=138 mmHg. A flow corrected PWV is calculate at diastolic bloodpressure (PWV_(f)=5.0 m/s−0.0 m/s=5.0 m/s) with the associated diastolicblood pressure determined from the array as p_(d)=84 mmHg. The arterialcompliance parameter distensibility (D) is determined from the systolicand diastolic PWV_(f) measures and the associated array elements. Inthis paper example the diastolic distensibility is identified as

${D = {57.6\frac{1}{MPa}}},$

and the systolic distensibility is identified as

$D = {31.3{\frac{1}{MPa}.}}$

What is claimed:
 1. A method for determining material characteristicsfor an artery, the method comprising: providing at least three valuesfor each of blood pressure, internal radius, and external radius of anarterial segment or segments of a subject; applying a base model offluid-structure interaction incorporating conservation of mass andmomentum for the fluid, and non-linear elasticity of the structure; andrunning a mathematical optimization on the base model to provide acalibrated model to determine material characteristics of the arterialsegment or segments.
 2. The method of claim 1, wherein the materialcharacteristics are a and c, R_(m), where a is a reduced materialconstant, ${a = {a_{11} - \frac{a_{12}^{2}}{a_{22}}}},$ a₁₁, a₁₂, a₂₂are the constants characterizing anisotropy of arterial wall, c is amaterial constant, and R_(m) is a mean wall radius in a load free state.3. The method of claim 1, wherein the mathematical optimization providesthe best fit for the equations${{{{ac}\; \lambda_{\theta \; k}^{2}e^{{aE}_{\theta \; k}^{2}}E_{\theta \; k}} - \frac{p_{k}{ri}_{k}}{h_{k}}} = 0},$k=1,2, . . . , N to determine a, c, and R_(m).
 4. A method fordetermining at least one of a blood pressure and an arterial complianceparameter of a subject, the method comprising; providing a value forpulse wave velocity within an arterial segment or segments of a subject;providing a value for flow velocity within the arterial segment orsegments of the subject; providing a value for blood density of thesubject; providing the material characteristics of an artery; andapplying a calibrated model of fluid-structure interaction incorporatingconservation of mass and momentum for the fluid, and non-linearelasticity of the structure, to calculate at least one of blood pressureand an arterial compliance parameter of the subject using the providedvalues.
 5. The method of claim 4, wherein the flow velocity is measureddirectly from the subject.
 6. The method of claim 4, wherein the flowvelocity is estimated based on PWV for the subject.
 7. The method ofclaim 4, wherein the blood pressure is systolic.
 8. The method of claim4, wherein the blood pressure is diastolic.
 9. The method of claim 4,wherein a peak pulse wave velocity is associated with a systolicpressure.
 10. The method of claim 4, wherein a minimum pulse wavevelocity is associated with a diastolic pressure.
 11. The method ofclaim 4, wherein a peak flow velocity is associated with a systolicblood pressure.
 12. The method of claim 4, wherein a minimum flowvelocity is associated with a diastolic blood pressure.